Conversion Between Implicit and Parametric Representation of Differential Varieties
Xiao-Shan Gao, Institute of Systems Science, Chinese Academy of Sciences.
Abstract: We will first give an introduction to the characteristic set method for algebraic differential equations, which could serve a basic tool for constructive algebraic geometry and provides a decision procedure for the first order theory of differential closed fields. As applications, we will present some recent results on conversion between implicit representation and parametric representation of differential varieties. For a set of differential rational parametric equations, we show how to compute its image and decide whether it represents a rational differential variety. The parametrization problem is much more difficult and is solved only in very limited cases. We give a polynomial time algorithm to find the rational parametrization for an autonomous first order ODE. Algorithms to find algebraic solutions for autonomous first order ODEs will also be discussed.